Richard E. Eykholt

Associate Professor
B.A., M.S., Ph.D., University of California - Irvine, 1978, 1980, 1984.
(970) 491-7366

Chaos and Nonlinear Dynamical Systems

Physicists are generally interested in studying the properties of materials and devices under increasingly more extreme conditions, in which case, the nonlinear properties of the material or device become more and more important. If the conditions become sufficiently extreme, (e.g., very strong driving or very high applied fields), the behavior of the system will finally become chaotic. Beyond this point, the chaotic response of the system makes it very difficult to study its properties.

However, recent research has shown that, in many cases, it may be possible to eliminate this chaotic behavior by making very small variations in the driving force or the applied field. Furthermore, the chaotic behavior may be replaced by any of a variety of steady-state or periodic behaviors. This ability to select from a variety of behaviors of the system may mean that the occurrence of chaos is actually an advantage. At present, chaos can be controlled in this manner only for very simple systems. However, we are working on ways to extend this approach to systems of much greater complexity (i.e., to the level of complexity found in real physical systems).

The other major focus of our research is the study of cellular automata (CA's). A CA is a discrete lattice in one or more dimensions whose sites may take on any of a discrete set of values. At discretized time steps, the values at the sites are updated, with the new value of a site depending on the old values of both that site and of neighboring sites. Hence, a CA is sort of a discretized version of a partial differential equation (PDE): in fact, a computer solves a PDE numerically by converting it to an equivalent CA.

Recently, researchers have begun to learn that CAs have greater flexibility in modeling physical systems than do PDEs, and this has led to an interest in the study of CAs as a modeling tool. Unfortunately, most of these studies have been numerical - very few analytic techniques exist for the study of CAs. On the other hand, we have recently developed an analytic method of solving for the exact equilibrium behavior of simple CAs (this is analogous to solving a PDE analytically). The presence of such an analytic tool for studying CAs would greatly aid research in this field. Hence, we are now studying this method to find out how widely applicable it is.

Our immediate goal is to understand which equilibrium behaviors can be found by this method and which, if any, cannot. If some behaviors cannot be found, we will investigate what properties they have in common in an effort to understand why they cannot be found. We then hope to generalize our method to allow it deal with more of these cases. At present, this method is fairly new, and this research is in its early stages.

These projects offer students in theoretical/mathematical physics a fairly even mix of analytical and numerical work. For the research on controlling chaos, the research itself is primarily analytical, but the procedures developed must then be demonstrated using numerical examples. For the research on cellular automata, the calculations using our new method are primarily analytical, although the equations sometimes become intractable and must be solved numerically. However, it is also necessary to perform numerical simulations of CAs to search for the various equilibrium behaviors that our method may not have found.